Matrix Analysis

1. Front Matter

   Pages i-xi

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2. A Review of Linear Algebra

   • Rajendra Bhatia

   Pages 1-27

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3. Majorisation and Doubly Stochastic Matrices

   • Rajendra Bhatia

   Pages 28-56

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4. Variational Principles for Eigenvalues

   • Rajendra Bhatia

   Pages 57-83

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5. Symmetric Norms

   • Rajendra Bhatia

   Pages 84-111

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6. Operator Monotone and Operator Convex Functions

   • Rajendra Bhatia

   Pages 112-151

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7. Spectral Variation of Normal Matrices

   • Rajendra Bhatia

   Pages 152-193

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8. Perturbation of Spectral Subspaces of Normal Matrices

   • Rajendra Bhatia

   Pages 194-225

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9. Spectral Variation of Nonnormal Matrices

   • Rajendra Bhatia

   Pages 226-252

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10. A Selection of Matrix Inequalities

    • Rajendra Bhatia

    Pages 253-288

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11. Perturbation of Matrix Functions

    • Rajendra Bhatia

    Pages 289-323

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12. Back Matter

    Pages 325-349

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A good part of matrix theory is functional analytic in spirit. This statement can be turned around. There are many problems in operator theory, where most of the complexities and subtleties are present in the finite-dimensional case. My purpose in writing this book is to present a systematic treatment of methods that are useful in the study of such problems. This book is intended for use as a text for upper division and gradu­ ate courses. Courses based on parts of the material have been given by me at the Indian Statistical Institute and at the University of Toronto (in collaboration with Chandler Davis). The book should also be useful as a reference for research workers in linear algebra, operator theory, mathe­ matical physics and numerical analysis. A possible subtitle of this book could be Matrix Inequalities. A reader who works through the book should expect to become proficient in the art of deriving such inequalities. Other authors have compared this art to that of cutting diamonds. One first has to acquire hard tools and then learn how to use them delicately. The reader is expected to be very thoroughly familiar with basic lin­ ear algebra. The standard texts Finite-Dimensional Vector Spaces by P.R.

• algebra
• approximation
• calculus
• Eigenvalue
• exponential function
• inequality
• linear algebra
• matrices
• matrix
• numerical analysis
• operator
• operator theory
• perturbation
• polynomial
• Smooth function

R. Bhatia

Matrix Analysis

"A highly readable and attractive account of the subject. The book is a must for anyone working in matrix analysis; it can be recommended to graduate students as well as to specialists."—ZENTRALBLATT MATH

"There is an ample selection of exercises carefully positioned throughout the text. In addition each chapter includes problems of varying difficulty in which themes from the main text are extended."—MATHEMATICAL REVIEWS

• Indian Statistical Institute, New Delhi, India

  Rajendra Bhatia